Joint mathematics project with a local junior school
John Mower - Senior Lecturer, University of Hertfordshire
Rationale for the project
The purpose of this project was to explore the use of practical resources in mathematics. The aims were to facilitate the learning of the pupils involved and to collate evidence and examples for sharing with student teachers. Part of my role as a University lecturer in primary mathematics is to stress the importance to students of the appropriate use of resources and to give relevant examples of how these have been effectively used in schools. This then allows us to link theory to practice effectively.
In undertaking this project I worked with staff at a local junior school to create a series of maths sessions using these practical resources. These sessions will be explained below with examples from session plans and responses from the pupils to this way of learning.
The junior school has, this summer, invested in a range of mathematical resources for all classes, with the view that all children will utilise these on a regular basis. It was agreed that I would work with the two Year 5 teachers, to plan how we would use these effectively, and then deliver sessions and note responses accordingly. The importance of using manipulatives to enhance mathematical understanding is explored in great depth in mathematics literature. Haylock and Cockurn's (2008) connections model, for example, which extols the benefits of making conceptual connections between language, symbols, practical experience and a range of representations, is used heavily in our discussions with students. Mike Askew, (2015) a prominent figure in maths pedagogy, commented in a recent presentation at a dyscalculia conference:
I was keen, initially, to agree between myself and the teachers a generic rationale for developing this 'insight' into the effective use of these resources, which would then underpin our subsequent planning and the teachers' discussions with other staff. Eventually, we agreed on the following set of criteria so that the resources, if used well, would:
- Help to develop a conceptual, rather than just a procedural understanding
- Help children to explain their thinking, with an emphasis on them talking to each other
- Develop and foster independence
- Challenge all children and dispel the feel that resources are only to be used by lower ability or younger children
- Promote an investigative mind-set within all lessons
- Help to develop connections between different aspects of Mathematics.
I was keen for the planned sessions to relate to the topics that the Year 5 teachers were planning for anyway, rather than to consider questions such as: 'How can we effectively use Numicon to help to meet the generic criteria?' (Listed above).
After some discussion, the lesson plan was written and the following aims were agreed.
- To be able to identify the relationships between unknown values
- To use reasoning skills to help to solve simple problems.
When teaching the first session with the year 5 children, I asked both classes what they thought were the main benefits of using resources and I was surprised at the level of response I received. Whilst both classes initially suggested a more simplistic response of 'they help us to work out an answer', when I probed them a little further I received considered and complex responses, such as those listed below:
- If you are a visual learner it helps you to see what is happening
- If you are a kinaesthetic learner it can help you physically work questions out
- It helps you to investigate and explore. You can see why
- You can see how numbers relate to each other
- It helps you to find different ways to work out answers.
After the discussion about why we use resources, cited above, I got the children to explore the Cuisenaire rods, using the question, 'what do you notice?' As suspected, this very open question needed some further prompting and probing to get them to openly discuss the relationship between the colours. Again though, as expected, this was limited to phrases such as 'two whites are the same as one red.'
My subsequent aim was to take the children away from barriers to creative thinking. All the children assumed that the white rod represented 'one', but when I introduced the idea that any of the rods could equal one, and got them to explore this concept, these barriers were lifted dramatically. With a little prompting, many of the children started to really challenge their thinking. Using the pedagogy of 'if I know this, what else do I know?' they quickly began to explore strategies with complex starting points. One pair started with the notion that the yellow rod (seven times the length of the single white) would have the value of 19.7. Very quickly they worked out that as soon as they could establish the value of the single white, finding other values were easy. With a little prompting and scaffolding, all children could then, at their own level, create different proportional values, given their original starting point.
Without knowing the class, and the values that had been instilled throughout the year, I was nevertheless impressed at how quickly the children were able to embrace the idea of removing traditional boundaries and norms, and creating something challenging and different. As they had so perceptively articulated at the start of the session, the manipulatives helped them to see how numbers related to each other, in both a visual and kinaesthetic manner. That the children both enjoyed and fully engaged in the session helped to underline the success of using the Cuisenaire rods as a central tool in investigational work.
After two weeks, I revisited the school for another planning session. I was keen to establish how the children had responded, in the interim, to further use of the manipulatives. The Year 5 teachers were both pleased at how the children were beginning to break down barriers to their learning, with continued use of the different resources, and were getting more confident at 'having a go' when presented with a very open task. One particular instance that they commented on was how, when exploring equivalences using the Cuisenaire rods, they quite naturally started making algebraic equations, such as 2B + 4R = 14, or 14 = 7R. The teachers noted how much more readily the children accepted the algebraic language, when using the rods to scaffold and illustrate their understanding.
When planning our 2nd session, we were conscious of wanting the children to explore a number of different resources, including using pictorial representations on paper. One of the reasons for this is that we did not want them to feel restricted by the manipulatives that happened to be in the box. The focus of the session was to focus on the equals sign, with the principal aim for the children to be able to clearly articulate and demonstrate the function of the equals sign. The initial question was designed to assess likely misconceptions, with the view of exploring these as the session developed. The children were asked to complete the missing number in the following equation: 10 + 4 = ? + 5. As expected, about 60% answered 14, with about 25% giving the correct answer of 9 and about 15% answering 19. We deliberately asked the children to complete this anonymously, to ease fears of children who might be apprehensive about getting this 'wrong'.
Subsequent activities explored the concept of equality, including the exploration of 'function machines' where the children needed to establish which function would allow for all equations to be correct. The purpose of these activities were to dispel the likely view, endorsed by the responses to the initial question, that Mathematics is purely about a question being asked, followed by an equals sign which then leads to the calculation of the answer.
The children were clearly engaged with these activities, but it was only when asked to start representing these equations using different representations, did many of them feel able to prove that the functions chosen enabled the equations to balance. It was very satisfying to see them openly explore and experiment with Cuisenaire rods, Numicon, bead strings and drawings to help with these justifications. Interestingly, whilst most children, at the end of the lesson, were able to then give the correct answer to the original equation set (10 + 4 =? + 5), several able children needed to show this with the available resources before they were totally satisfied.
Our final session focused on problem solving, with the principal aim of getting the children to see the value of the use of resources when faced with a problem. Using counters to represent values rather than Cuisenaire rods forced the children to consider a more abstract representation, rather than the more physical values of the rods. We wondered whether the children would struggle with this more abstract concept, but this didn't seem to be the case, which was gratifying. The final activity, 'square circles' was a practical activity which posed the problem of why square numbers have an odd number of multiples, whereas all other numbers have an even amount. They were encouraged to use a range of resources, including pencil and paper, to explain this. The creativity showed in the range of approaches, as well as the confidence to try different resources was, again, very gratifying and left us all very satisfied that the project had succeeded in ones of its aims: to get the children to use different resources with both confidence and creativity.
After the final session we asked the children for comments on how the range of new resources had impacted on their learning. All of the children had enjoyed the sessions and felt that the resources had impacted on their learning. Again, there were some insightful comments. Phrases such as 'helped to understand why..', 'allowed you to challenge yourself' or 'helped me to understand rules/values' suggested that our project was largely successful in its principle aim of developing conceptual understanding, as well as procedural and factual.
I hope this project will help to kick-start the effective use of these resources at the school. From my perspective, I wanted to be able to give relevant and recent examples to my students which showed how a range of resources have helped children to understand, challenge themselves and heighten their enjoyment of Mathematics. I can now confidently share a range of these experiences with all of our students.
- Haylock, D. & Cockburn, A. (2008) Understanding Mathematics for Young Children, London: Sage.
- Askew, M. (2015) 'Promoting inclusive mathematics classrooms', conference paper presented at Dyscalculia and Maths Learning Difficulties Conference, London, 25 June.
LINK 2016, vol. 2, issue 1 / Copyright 2016 University of Hertfordshire